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Theorem eluniab 3712
Description: Membership in union of a class abstraction. (Contributed by NM, 11-Aug-1994.) (Revised by Mario Carneiro, 14-Nov-2016.)
Assertion
Ref Expression
eluniab (𝐴 {𝑥𝜑} ↔ ∃𝑥(𝐴𝑥𝜑))
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem eluniab
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 eluni 3703 . 2 (𝐴 {𝑥𝜑} ↔ ∃𝑦(𝐴𝑦𝑦 ∈ {𝑥𝜑}))
2 nfv 1489 . . . 4 𝑥 𝐴𝑦
3 nfsab1 2103 . . . 4 𝑥 𝑦 ∈ {𝑥𝜑}
42, 3nfan 1525 . . 3 𝑥(𝐴𝑦𝑦 ∈ {𝑥𝜑})
5 nfv 1489 . . 3 𝑦(𝐴𝑥𝜑)
6 eleq2 2176 . . . 4 (𝑦 = 𝑥 → (𝐴𝑦𝐴𝑥))
7 eleq1 2175 . . . . 5 (𝑦 = 𝑥 → (𝑦 ∈ {𝑥𝜑} ↔ 𝑥 ∈ {𝑥𝜑}))
8 abid 2101 . . . . 5 (𝑥 ∈ {𝑥𝜑} ↔ 𝜑)
97, 8syl6bb 195 . . . 4 (𝑦 = 𝑥 → (𝑦 ∈ {𝑥𝜑} ↔ 𝜑))
106, 9anbi12d 462 . . 3 (𝑦 = 𝑥 → ((𝐴𝑦𝑦 ∈ {𝑥𝜑}) ↔ (𝐴𝑥𝜑)))
114, 5, 10cbvex 1710 . 2 (∃𝑦(𝐴𝑦𝑦 ∈ {𝑥𝜑}) ↔ ∃𝑥(𝐴𝑥𝜑))
121, 11bitri 183 1 (𝐴 {𝑥𝜑} ↔ ∃𝑥(𝐴𝑥𝜑))
Colors of variables: wff set class
Syntax hints:  wa 103  wb 104  wex 1449  wcel 1461  {cab 2099   cuni 3700
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095
This theorem depends on definitions:  df-bi 116  df-tru 1315  df-nf 1418  df-sb 1717  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-v 2657  df-uni 3701
This theorem is referenced by:  elunirab  3713  dfiun2g  3809  inuni  4038  snnex  4327  elfv  5371  unielxp  6024  tfrlem9  6168  tfr0dm  6171  metrest  12489
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